Counting Eights: An Introduction to Probability

**1.
Warm-Up**

Roll a pair of dice and record
the **sum** of the dots showing on the dice ________

**2. Estimating the
Probability of Rolling an 8**

(A) In Minitab, give column C1
the name **Trial**

*Task*: Make C1
consist of the whole numbers from 1 to 1000.

*Select*:
Calc > Make Patterned Data > Simple Set of Numbers >

From first value =1 to last value = 1000

(B) Give column C2 the name
**Red Die**

*Task*: Make C2
consist of 1000 rolls of a balanced die

*Select*: Calc
> Random Data > Integer >

Minimum value = 1 and Maximum value = 6

(C) Give column C3 the name
**Green Die**

*Task*: Make C3
consist of 1000 rolls of another balanced die

*Select*: Calc
> Random Data > Integer >

Minimum value = 1 and Maximum value = 6

(D) Give column C4 the name
**Sum of Dice**

*Task*: Make C4
the sum of Red Die and Green Die

*Select*: Calc
> Calculator

(E) Give column C5 the name
**Sum=8?**

*Task*: Make C5
have value 1 if Sum of Dice is 8, and a 0 if Sum of Dice is not 8.

*Select*: Calc
> Calculator, and now type (C4=8) in the Expression box. Minitab
will

recognize this as a logical expression and place a 1 where the expression is true and place

a 0 where the expression is false.

*Check to make sure your commands worked as
desired.*

(F) Give column C6 the
name **8s So Far**

*Task*: Make C6
the number of 8s rolled so far in your sequence of rolls

for partial sums.

*Check to make sure your commands worked as
desired.*

(G) Give column C7 the name
**Proportion 8s So Far**

*Task*: Make C7
the proportion of your rolls so far that have been an 8

*Select*: Calc
> Calculator, and now type C6/C1 in the Expression box.

(H) What is **your** value
of **Proportion 8s So Far** at trial number 10?
________

What is **your** value of
**Proportion 8s So Far** at trial number 25?
________

What is **your** value of
**Proportion 8s So Far** at trial number 50?
________

What is **your** value of
**Proportion 8s So Far** at trial number 100?
________

What is **your** value of
**Proportion 8s So Far** at trial number 500?
________

What is **your** value of
**Proportion 8s So Far** at trial number 1000?
________

Place **dots** on the board
at the appropriate location for **your** six numbers above.

- Make a dotplot for the student values of the variable
*Y*at*n*= 10, 25, 50, 100, 500, and 1000 rolls. For each dotplot, describe the center, spread, and shape of the distribution. Now looking over the six dotplots, which distributional characteristics seem to stay constant from plot to plot, and which characteristics appear to be changing from plot to plot? If a characteristic is changing, explain how it is changing.

- Plot your 1000 values of
*Y*versus roll number*n*. Comment on the appearance of the graph and any noticeable trends.

- Consider all the possible
pairs of values (red die, green die) when balanced red and green dice are
rolled. Argue why there are 36 equally likely such pairs and use this
fact to find
_{}, the exact theoretical probability of rolling a sum of 8 when two dice are rolled.

- In problem 3 you found the
value of
_{}, the exact probability of an 8 when two balanced dice are rolled. Look again at the*Y*values of the students in your class. For each value of*n*= 10, 25, 50, 100, 500, and 1000, what proportion of student*Y*values

(a) fall within 0.05 of _{}?

(b) fall within 0.03 of _{}?

(c) fall within 0.01 of _{}?

How
does the proportion of *Y* values falling within a specified range from
_{}

depend on
*n*? For a fixed small quantity *d*, what do you think will
happen to the

proportion
of student *Y* values within *d *of _{} as the
number of rolls *n* gets

increasingly larger?

- If we think of
*Y*as an estimate of the probability_{}, then the relative error of estimation would be

_{}.

For *n* = 10, 25, 50, 100, 500, and 1000, what
proportion of student *Y* values

provide a relative error of estimation of less than 0.10, or 10%? How does

this proportion change with *n*?

- Compare your graph in question 2 with as many of your fellow students as

possible. Keep in mind that you all have *different*
random simulations and

therefore very different sequences of 1000 rolls. Which regions of your graph are

similar to those of other students, and which regions of your graph differ from

those of other students? After your comparisons, complete this sentence: The

most striking characteristic that is *common* to my and
my fellow students’ graphs

is _________ . Can you state this *apparent
law* of random behavior in a concise

statement?

- Find the sample standard
deviation,
_{}, of the student*Y*values at*n*= 10, 25, 50, 100, 500, and 1000.

(a) Plot the six points _{}on a graph,
where_{}denotes the natural or base *e* logarithm. Describe
the appearance of this plot.

(b) Find the intercept _{}and slope
_{} of the least squares regression line through the six points
graphed in (a).

(c) Show that (a) and (b) imply that
_{}, where _{} is a
constant. It turns out the *theoretical* values of _{} and
_{} are _{} and _{}.
Compare these theoretical values with your estimates.